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# Full text of "Treatise On Analysis Vol-Ii"

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```208      XIII   INTEGRATION

D of points x 6 X such that \n\({x}) > 0 (13.18.5). This set is clearly //-meas-
urable, and if we write n = h'\n\ (13.16.3), the measure /j,2 = (<pDh) - \n\ is
concentrated on D (13.14.1), and ^1=^-^2 is diffuse. For we have
102 = <PD*M ((13.13.4) and (13.16.3)), hence |^| = |/x| - |/*2|; for each
xeD, we have \fi\({x}) = |ju2 !({*}) by construction, and for each xtf-D
we have \fi\({x}) = |/£2|({x}) = ° bY the definition of D; hence, by (13.16.1),

(13.18.7) Let 1 be Lebesgue measure on R. From (13.18.4) and (13.18.6) it
follows that every measure /^ on R is uniquely of the form ^ + p2 + /j3,
where ^ = # • A is a measure with base A, /i2 is an atomic measure and ju3 is a
diffuse measure disjoint from A and therefore concentrated on a A-negligible
set (necessarily nondenumerable).

Remarks

(13.18.8) (i) If ju is an atomic measure, the denumerable set of points
x € X such that |/x|({jc}) > 0 is the smallest set on which // is concentrated, On
the other hand, for a diffuse measure v ^ 0, there exists no smallest set on
which v is concentrated: in other words, there exists no largest v-negligible set,
because every set consisting of a single point is v-negligible.

(ii) Let n be an atomic measure concentrated on a denumerable set D,
and let (an) be the sequence consisting of the distinct points of D arranged in
some order. Put \n\({an}) = yn > 0. If Dn = [ai9 . . . , «„}, then cpD = sup <pDro;

n

since \n\ = <pD • |^|, then by (13.14.1) and (13.5.7) we have

for any function /^O (the sum on the right is either a finite real number

or +00). This implies in particular that, for any compact set K, we have

X yn < +00. Furthermore, since every set consisting of a single point is

<JMeK

//-measurable and since X - D is /(-negligible, it follows that every mapping
of X into a topological space is ju-measurable. The /i-integrable functions

00

are therefore those for which £ yJ/(OI < +00 (13.9.13), and we havetely
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